# American Institute of Mathematical Sciences

February  2015, 35(2): 703-723. doi: 10.3934/dcds.2015.35.703

## Unbounded perturbations of the generator domain

 1 Department of Mathematics, Faculty of Sciences, Ibn Zohr University, B.P. 8106, Agadir, Morocco 2 Department of Information Engineering and Applied Mathematics, University of Salerno, via Ponte Don Melillo, 84084 Fisciano (SA), Italy 3 Dept. of Information Eng., Electrical Eng. and Applied Mathematics, University of Salerno, Via Giovanni Paolo II, 132, I 84084 Fisciano (SA), Italy

Received  January 2013 Revised  January 2014 Published  September 2014

Let $X,U$ and $Z$ be Banach spaces such that $Z\subset X$ (with continuous and dense embedding), $L:Z\to X$ be a closed linear operator and consider closed linear operators $G,M:Z\to U$. Putting conditions on $G$ and $M$ we show that the operator $\mathcal{A}=L$ with domain $D(\mathcal{A})=\big\{z\in Z:Gz=Mz\big\}$ generates a $C_0$-semigroup on $X$. Moreover, we give a variation of constants formula for the solution of the following inhomogeneous problem \begin{align*} \begin{cases} \dot{z}(t)=L z(t)+f(t),& t\ge 0,\cr G z(t)=Mz(t)+g(t),& t\ge 0,\cr z(0)=z^0. \end{cases} \end{align*} Several examples will be given, in particular a heat equation with distributed unbounded delay at the boundary condition.
Citation: Said Hadd, Rosanna Manzo, Abdelaziz Rhandi. Unbounded perturbations of the generator domain. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 703-723. doi: 10.3934/dcds.2015.35.703
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